Provedalphabaguettenoether-checksumreview-2

Randomizable pairing: Σᵢ⟨dQᵢ, M·qᵢ⟩ = Σⱼ⟨dKⱼ, Mᵀ·kⱼ⟩ for EVERY D×D matrix M

For all $\htmlData{mx=1}{T, D \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=9}{R : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{T}} \mathrel{\htmlData{mx=8}{\to}} \htmlData{mx=5}{\htmlData{mx=3}{\operatorname{Fin}}\,\htmlData{mx=4}{T}} \mathrel{\htmlData{mx=7}{\to}} \htmlData{mx=6}{\mathbb{R}}}$ , $\htmlData{mx=9}{q, k : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{T}} \mathrel{\htmlData{mx=8}{\to}} \htmlData{mx=5}{\htmlData{mx=3}{\operatorname{Fin}}\,\htmlData{mx=4}{D}} \mathrel{\htmlData{mx=7}{\to}} \htmlData{mx=6}{\mathbb{R}}}$ , $\htmlData{mx=9}{M : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{D}} \mathrel{\htmlData{mx=8}{\to}} \htmlData{mx=5}{\htmlData{mx=3}{\operatorname{Fin}}\,\htmlData{mx=4}{D}} \mathrel{\htmlData{mx=7}{\to}} \htmlData{mx=6}{\mathbb{R}}}$ and $\htmlData{mx=1}{\mathit{scale} \in \htmlData{mx=0}{\mathbb{R}}}$ :

\begin{aligned} & \htmlData{mx=25}{\sum_{i} \htmlData{mx=24}{\sum_{a} \htmlData{mx=23}{\htmlData{mx=12}{\htmlData{mx=1}{\mathit{scale}} \cdot \htmlData{mx=11}{\sum_{j} \htmlData{mx=10}{\htmlData{mx=5}{\htmlData{mx=2}{R}(\htmlData{mx=3}{i}, \htmlData{mx=4}{j})} \cdot \htmlData{mx=9}{\htmlData{mx=6}{k}(\htmlData{mx=7}{j}, \htmlData{mx=8}{a})}}}} \cdot \left(\htmlData{mx=22}{\sum_{b} \htmlData{mx=21}{\htmlData{mx=16}{\htmlData{mx=13}{M}(\htmlData{mx=14}{a}, \htmlData{mx=15}{b})} \cdot \htmlData{mx=20}{\htmlData{mx=17}{q}(\htmlData{mx=18}{i}, \htmlData{mx=19}{b})}}}\right)}}} \\ & \quad \htmlData{mx=0}{=} \htmlData{mx=50}{\sum_{j} \htmlData{mx=49}{\sum_{b} \htmlData{mx=48}{\htmlData{mx=37}{\htmlData{mx=26}{\mathit{scale}} \cdot \htmlData{mx=36}{\sum_{i} \htmlData{mx=35}{\htmlData{mx=30}{\htmlData{mx=27}{R}(\htmlData{mx=28}{i}, \htmlData{mx=29}{j})} \cdot \htmlData{mx=34}{\htmlData{mx=31}{q}(\htmlData{mx=32}{i}, \htmlData{mx=33}{b})}}}} \cdot \left(\htmlData{mx=47}{\sum_{a} \htmlData{mx=46}{\htmlData{mx=41}{\htmlData{mx=38}{M}(\htmlData{mx=39}{a}, \htmlData{mx=40}{b})} \cdot \htmlData{mx=45}{\htmlData{mx=42}{k}(\htmlData{mx=43}{j}, \htmlData{mx=44}{a})}}}\right)}}} \end{aligned}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
x ∈ TVariables: T, D range over natural numbers: the statement must hold for every choice. ((T D : ℕ))
Fin: Fin: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fin)
f(x)Function application: Fin applied to T. Lean writes f x; papers write f(x). (Fin T)
x : TVariables: q, k range over Fin T → Fin D → ℝ: the statement must hold for every choice. ((q k : Fin T → Fin D → ℝ))
=Equality: ∑ i, ∑ a, scale · (∑ j, R(i, j) · k(j, a)) · (∑ b, M(a, b)… and ∑ j, ∑ b, scale · (∑ i, R(i, j) · q(i, b)) · (∑ a, M(a, b)… are the same. (∑ i, ∑ a, (scale * ∑ j, R i j * k j a) …)
⟹Implies: If Fin(T), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin T → ℝ)
RReal numbers: The real numbers: every point on the number line. (ℝ)
∑Summation: Add up R(i, j) · k(j, a) for every j. (∑ j, R i j * k j a)
·Multiplication: R(i, j) times k(j, a). (R i j * k j a)

theorem nc2-pairing-holds-for-any-matrix {T D : ℕ} (R : Fin T → Fin T → ℝ) (q k : Fin T → Fin D → ℝ) (M : Fin D → Fin D → ℝ) (scale : ℝ) :
    ∑ i, ∑ a, (scale * ∑ j, R i j * k j a) * (∑ b, M a b * q i b) = ∑ j, ∑ b, (scale * ∑ i, R i j * q i b) * (∑ a, M a b * k j a)

Generalizes C3 (M = identity) to a whole family of checks. For any R and any M, both sides equal scale·Σᵢⱼ Rᵢⱼ⟨kⱼ, M qᵢ⟩. Drawing M at random per step makes the check unpredictable to a structured bug; computing M·q is one 64×64 GEMM per head. Part of 'Noether checksums for Baguette's backward kernels' (review pass 2).

How it was proved

Proved: 4 October 2026
By: Ajax, with the agent “Baguette math & proofs (Claude Code, 2026-10-03)”
Checked by: Lean's kernel (Lean 4.33.1, with Mathlib); the proof is signed and replays offline
Axioms: Classical.choice, Quot.sound, propext
Proof: 32 lines of Lean

Randomizable pairing: Σᵢ⟨dQᵢ, M·qᵢ⟩ = Σⱼ⟨dKⱼ, Mᵀ·kⱼ⟩ for EVERY D×D matrix M

How it was proved

More proved statements